
\(\mathrm {(Non}\) -\(\mathrm {resonant\ case)}\)  If for any \( n\in {\mathbb {Z}}^d \)  with \(0<|n|\le N \) ,
\(|2\xi -\langle n,\alpha \rangle |_{\ge \epsilon ^{\sigma },\ }where  |\cdot |_{=\inf _{j\in {\mathbb {Z}}}|\cdot -j| , then we have  \ \Vert A_+-A\Vert \le 4\Vert A\Vert \epsilon ,\ |f_+|_{r_+}\le \epsilon ^{2},\ |B-\mathrm {Id}|_{r_+}\le 2\epsilon ^{\frac{1}{2}}. \item (\mathrm {Resonant\ case}) If there exists  n_*\in {\mathbb {Z}}^d  with  0<|n_*|\le N  such that\begin{equation}|2\xi -\langle n_*,\alpha \rangle |_{<\epsilon ^{\sigma },} then we have A_+=e^{a_+} with \Vert a_+\Vert \le 4\epsilon ^{\sigma }, and{\begin{@align*}{1}{-1}&|f_+|_{r_+}\le \epsilon ^{\frac{3}{4}-\frac{2 r_+}{r-r_+}} e^{-(r-r_+) \epsilon ^{-\frac{10\nu }{11}\frac{\sigma }{\tau }}}\ll \epsilon ^{2}, \\&|B|_{r^{\prime \prime }}\le 32 \gamma ^{-\frac{1}{2}}\Vert A\Vert ^{\frac{1}{2}} |n_*|^{\frac{\tau }{2}}e^{r^{\prime \prime }(\pi |n_*|)^{\nu }},\ \forall r^{\prime \prime }\in (0, r_+],\end{@align*}}and \deg B=n_*. Moreover, let  a_+=M^{-1}\begin{pmatrix}\mathrm {i}t_+ & \mu _+\\\overline{\mu _+} & -\mathrm {i}t_+\end{pmatrix}M , we have\begin{equation} |t_+|\le \epsilon ^{\sigma }, |\mu _+|\le \epsilon ^{\frac{3}{4}}e^{-r|2\pi n_*|^{\nu }}.\end{equation}\end{equation}}\) 
For \(\mathrm {I}\) . Let \(\eta =2\Vert A\Vert ^2\epsilon ^{\frac{1}{9}}\) , then we have the corresponding decomposition, by (REF ), we can apply Lemma REF , then \( (\alpha ,Ae^f) \)  is conjugated to a cocycle \( (\alpha ,Ae^g) \)  with \( g\in G_{r}^{\nu \,(re)}(\eta ) \) .
The fact \( \alpha \in \mathrm {DC}_d(\gamma ,\tau ) \)  and (REF ) can imply
\(|e^{\mathrm {i}2\pi \langle n,\alpha \rangle }-1|\ge \frac{\gamma }{N^\tau }\ge \epsilon ^{\sigma }\ge \eta ,\qquad \forall n\in {\mathbb {Z}}^d \text{ with } 0<|n|\le N. \) 
Thus combining (REF ) with (REF ), by the definition of \( G_{r}^{\nu \,(re)}(\eta ) \) , we have
\( \mathcal {T}_N g(\theta )-\widehat{g}(0)=0,\) 
hence
\(|g-\widehat{g}(0)|_{r_+}=\sum _{|n|>N}|\widehat{g}(n)|e^{ r_+ (2\pi |n|)^{\nu }}\le |g|_r e^{-(r-r_+)(2\pi N)^{\nu }}\le 2\epsilon ^{3}.\) 

Let \( B:=e^Y\) , \(A_+:=Ae^{\widehat{g}(0)} \) , and \( e^{f_+}:=e^{-\widehat{g}(0)}e^{g} \) . Clearly, we have the corresponding estimates.
For \( \mathrm {II} \) . Since \( \alpha \in \mathrm {DC}_d(\gamma ,\tau ) \) , thus we only need to consider the case in which \( A \)  is elliptic, i.e., \( \xi \in {\mathbb {R}}\backslash \lbrace 0\rbrace  \) . In fact, if \(\xi \in \mathrm {i}{\mathbb {R}}\) , we must have (REF ) holds.
Lemma 4.1 For \( A \)  is elliptic, there exists \(P\in \mathrm {SL}(2,{\mathbb {R}}) \)  such that
\(A=P^{-1}M^{-1}\begin{pmatrix}e^{\mathrm {i}2\pi \xi }&0\\0&e^{-\mathrm {i}2\pi \xi }\end{pmatrix}MP\) 
with \(\Vert P\Vert \le 16\max \lbrace \frac{1}{2},\sqrt{\frac{\Vert A\Vert }{2|\sin 2\pi \xi |}}\rbrace \) .

By Schur's Lemma, there exists a unitary matrix \(U\)  such that
\(A=e^{a}=e^{U\begin{pmatrix}\mathrm {i}2\pi \xi &p\\0&-\mathrm {i}2\pi \xi \end{pmatrix}U^*}=U\begin{pmatrix}e^{\mathrm {i}2\pi \xi }&\frac{\sin 2\pi \xi }{2\pi \xi }p\\0&e^{-\mathrm {i}2\pi \xi }\end{pmatrix}U^*.\) 
Notice that \( \Vert U\Vert \le 2 \) , we have \( \Vert a \Vert \le 8\max \lbrace |2\pi \xi |,|p|\rbrace  \) , and \( |\frac{\sin 2\pi \xi }{2\pi \xi }||p|\le 4\Vert A\Vert \) , then by Lemma 8.1 in [1]}, there exists \( P\in \mathrm {SL}(2,{\mathbb {R}}) \)  with \( \Vert P\Vert \le 2\sqrt{\frac{\Vert a\Vert }{|2\pi \xi |}} \)  such that \( P a P^{-1}=\begin{pmatrix}0&2\pi \xi \\-2\pi \xi &0\end{pmatrix} \) , then we have
\(PAP^{-1}=M^{-1}\begin{pmatrix}e^{\mathrm {i}2\pi \xi }&0\\0&e^{-\mathrm {i}2\pi \xi }\end{pmatrix}M\) 
with \( \Vert P\Vert \le 16\max \lbrace \frac{1}{2},\sqrt{\frac{\Vert A\Vert }{2|\sin 2\pi \xi |}}\rbrace  \) .
Let \( \tilde{A}:=M^{-1}\begin{pmatrix}e^{\mathrm {i}2\pi \xi }&0\\0&e^{-\mathrm {i}2\pi \xi }\end{pmatrix}M, \tilde{f}:=PfP^{-1} \) , by (REF ) and (),
\(|2\xi |_{\ge |\langle n_*,\alpha \rangle |_{-\epsilon ^\sigma \ge \frac{\gamma }{|n_*|^\tau }-\epsilon ^\sigma \ge \frac{\gamma }{2|n_*|^\tau },then by  |\sin 2\pi \xi |\ge \frac{2}{\pi }|2\xi |_{ , we have|\tilde{f}|_r \le \Vert P\Vert ^2\epsilon :=\tilde{\epsilon }\le 256\max \lbrace \frac{1}{4},\gamma ^{-1}\Vert A\Vert |n_*|^{\tau }\rbrace \epsilon \le 256\gamma ^{-1}\Vert A\Vert |n_*|^{\tau }\epsilon \le \epsilon ^{\frac{8}{9}}.By the concrete algebraic structure of \tilde{A}, if we set\begin{aligned}&\Phi _1:=\lbrace n\in {\mathbb {Z}}^d:|e^{\mathrm {i}2\pi \langle n,\alpha \rangle }-1|\ge \eta \rbrace ,\\&\Phi _2:=\lbrace n\in {\mathbb {Z}}^d:|e^{\mathrm {i}2\pi (2\xi -\langle n,\alpha \rangle )}-1|\ge \eta \rbrace ,\end{aligned}and redefine{\begin{@align*}{1}{-1}\widetilde{G}_{r}^{\nu \,(nre)}(\eta )&=\left\lbrace \left.\begin{pmatrix}\mathrm {i}\mathcal {S}_{\Phi _1}f&\mathcal {S}_{\Phi _2} g\\\mathcal {S}_{-\Phi _2} \bar{g}&-\mathrm {i}\mathcal {S}_{\Phi _1}f\end{pmatrix}\right|f\in G^{\nu }_{r}(\mathbb {T}^d,{\mathbb {R}}), g\in G^{\nu }_{r}(\mathbb {T}^d,\mathbb {C})\right\rbrace ,\\\widetilde{G}_{r}^{\nu \,(re)}(\eta )&=\left\lbrace \left.\begin{pmatrix}\mathrm {i}\mathcal {S}_{\Phi _1^c}f&\mathcal {S}_{\Phi _2^c} g\\\mathcal {S}_{-\Phi _2^c} \bar{g}&-\mathrm {i}\mathcal {S}_{\Phi _1^c}f\end{pmatrix}\right|f\in G^{\nu }_{r}(\mathbb {T}^d,{\mathbb {R}}), g\in G^{\nu }_{r}(\mathbb {T}^d,\mathbb {C})\right\rbrace ,\end{@align*}}G_{r}^{\nu \,(nre)}(\eta ), G_{r}^{\nu \,(re)}(\eta ) are redefined similar, we get the corresponding decomposition. Then we have the following which is similar with Lemma \ref {lem3.1}.\begin{lemma}Assume  A\in \mathrm {SO(2,{\mathbb {R}})} , there exist numerical constants  c_2,D_2  such that if \epsilon \le c_2\Vert A\Vert ^{-D_2} \ \text{and}\ \ \eta \ge \Vert A\Vert ^2\epsilon ^{\frac{1}{3}},then for any  f\in G^{\nu }_{r}(\mathbb {T}^d,\mathrm {sl(2,{\mathbb {R}})})  with  |f|_{r}\le \epsilon , there exist  Y\in G_{r}^{\nu \,(nre)}(\eta )  and  f^{(re)}\in G_{r}^{\nu \,(re)}(\eta )  such that  e^{-Y(\theta +\alpha )}(Ae^{f(\theta )})e^{Y(\theta )}=Ae^{f^{(re)}(\theta )},with  |Y|_{r}\le \epsilon ^{\frac{1}{2}} , and  |f^{(re)}|_{r}\le 2\epsilon .\end{lemma}}{\begin{xmlelement*}{proof}The only difference between Lemma \ref {lem3.1} and Lemma \ref {lem2.2} is that if A\in SO(2,{\mathbb {R}}), for any Y\in \mathrm {sl}(2,{\mathbb {R}}) , A^{-1}Y(\cdot +\alpha )A-Y(\cdot ) still belongs to \mathrm {sl}(2,{\mathbb {R}}). By the definition of redefined non-resonant space, one can easily check that|A^{-1}Y^{\prime }(\theta +\alpha )A-Y^{\prime }(\theta )|_r\ge \eta |Y^{\prime }|_r.The rest follows a same line as Lemma \ref {lem2.2}.\end{xmlelement*}}}Let  \eta :=2\Vert \tilde{A}\Vert ^2\tilde{\epsilon }^\frac{1}{3} , then one can apply Lemma \ref {lem2.2}, thus (\alpha ,\tilde{A}e^{\tilde{f}}) can be conjugated to (\alpha ,\tilde{A}e^g) with g\in G_{r}^{\nu \,(re)}(\eta ) by a conjugacy e^Y, and |Y|_r\le \tilde{\epsilon }^{\frac{1}{2}} , |g|_r\le 2\tilde{\epsilon }.By the definition of G_{r}^{\nu \,(re)}(\eta ), there exist  g_{11}(\theta )\in G^\nu _{r}(\mathbb {T}^d,{\mathbb {R}}) ,  g_{12}(\theta )\in G^\nu _{r}(\mathbb {T}^d,\mathbb {C})  such that{\begin{@align}{1}{-1}Mg(\theta )M^{-1} &=\begin{pmatrix}\mathrm {i}\mathcal {S}_{\Phi _1^c} g_{11}(\theta ),\mathcal {S}_{\Phi _2^c} g_{12}(\theta )\\\mathcal {S}_{-\Phi _2^c} \overline{g_{12}}(\theta ),-\mathrm {i}\mathcal {S}_{\Phi _1^c} g_{11}(\theta )\end{pmatrix}\\&=\sum _{n\in \Phi _1^c}\begin{pmatrix}\mathrm {i} \widehat{g_{11}}(n)&0\\0&-\mathrm {i}\widehat{g_{11}}(n)\end{pmatrix}e^{\mathrm {i}2\pi \langle n,\theta \rangle }\\&+\sum _{n\in \Phi _2^c}\begin{pmatrix}0&\widehat{g_{12}}(n)e^{\mathrm {i}2\pi \langle n,\theta \rangle }\\ \overline{\widehat{g_{12}}(n)}e^{-\mathrm {i}2\pi \langle n,\theta \rangle }&0\end{pmatrix}.\end{@align}}}\) 
By the Diophantine condition and (), we have the following observation:
Claim 1
\(\Phi _1^c\cap \lbrace n\in {\mathbb {Z}}^d:|n|\le N\rbrace =\lbrace 0\rbrace ,\Phi _2^c\cap \lbrace n\in {\mathbb {Z}}^d:|n|\le N\rbrace =\lbrace n_*\rbrace .\) 

For any \(n\ne 0\)  and \(n\in \Phi _1^c\) , we have
\(\frac{\gamma }{|n|^\tau }\le |e^{\mathrm {i}2\pi \langle n,\alpha \rangle }-1|<\eta \le \epsilon ^\sigma , \) 
thus \(|n|> \gamma ^{\frac{1}{\tau }}\epsilon ^{-\frac{\sigma }{\tau }}.\)  For any \( n\in \Phi _2^c \) , we have
\(|2\xi -\langle n,\alpha \rangle |_{<\frac{1}{4}\eta . Assume there is an  n_*^{\prime }\ne n_*  with  |2\xi -\langle n_* ^{\prime },\alpha \rangle |_{ <\frac{1}{4}\eta , then we have\frac{\gamma }{|n_*^{\prime }-n_*|^\tau }\le |e^{\mathrm {i}2\pi \langle n_*^{\prime }-n_*,\alpha \rangle }-1|\le 2|\langle n_*^{\prime }-n_*, \alpha \rangle |_{<2\epsilon ^{\sigma },which implies  |n_*^{\prime }|>2^{-\frac{1}{\tau }}\gamma ^{\frac{1}{\tau }}\epsilon ^{-\frac{\sigma }{\tau }}-N .Let  N_1:=\gamma ^{\frac{1}{\tau }}\epsilon ^{-\frac{\sigma }{\tau }} ,  N_2:=2^{-\frac{1}{\tau }}\gamma ^{\frac{1}{\tau }}\epsilon ^{-\frac{\sigma }{\tau }}-N , then by (\ref {epsilon}), we have \epsilon ^{-\frac{10\sigma }{11\tau }}\le 2^{-1-\frac{1}{\tau }}\gamma ^{\frac{1}{\tau }}\epsilon ^{-\frac{\sigma }{\tau }} and we get N_1>N_2\gg N.}By the Claim \ref {claim}, we have{\begin{@align}{1}{-1}Mg(\theta )M^{-1}=&\begin{pmatrix}\mathrm {i} \widehat{g_{11}}(0)&\widehat{g_{12}}(n_*)e^{\mathrm {i}2\pi \langle n_*,\theta \rangle }\\ \overline{\widehat{g_{12}}(n_*)}e^{-\mathrm {i}2\pi \langle n_*,\theta \rangle }&-\mathrm {i}\widehat{g_{11}}(0)\end{pmatrix}+\mathcal {R}_g(\theta ).\end{@align}}Hence we have|\mathcal {R}_g|_{r_+}\le 2 \tilde{\epsilon }e^{-(r-r_+)(2\pi N_2)^{\nu }}\le 2\epsilon ^{\frac{8}{9}} e^{-(r-r_+)(2\pi N_2)^{\nu }}.}}\) 
Now we define a \((2{{\mathbb {Z}}})^d\) -periodic rotation
\(Z(\theta ):=e^{\pi {\langle n_*,\theta \rangle }J}=M^{-1}\begin{pmatrix}e^{\frac{\mathrm {i}2\pi \langle n_*,\theta \rangle }{2}}&0\\0&e^{-\frac{\mathrm {i}2\pi \langle n_*,\theta \rangle }{2}}\end{pmatrix}M,\) 
where \( J \)  is the symplectic matrix \(\begin{pmatrix}0&1\\-1&0\end{pmatrix}\) . Obviously, \( Z\in G^{\nu }_{r}(2\mathbb {T}^d, \mathrm {SL}(2,{\mathbb {R}})) \) , and for any \( r^{\prime \prime }\in (0,r) \) , \( |Z|_{r^{\prime \prime }}\le 2e^{r^{\prime \prime }(\pi |n_*|)^{\nu }} \) . Given any \( g\in G_{r}^{\nu \ (re)}(\eta ) \) , we have
\( Z(\theta +\alpha )^{-1}(\tilde{ A}e^g)Z(\theta )=M^{-1}\begin{pmatrix}e^{\mathrm {i}2\pi (\xi -\frac{\langle n_*,\alpha \rangle }{2})}&0\\0&e^{-\mathrm {i}2\pi (\xi -\frac{\langle n_*,\alpha \rangle }{2})}\end{pmatrix}Me^{Z^{-1}(\theta )gZ(\theta )}. \) 
By a direct calculation, we get
\( Z(\theta )^{-1}g(\theta )Z(\theta )= M^{-1}\begin{pmatrix}\mathrm {i}\widehat{g_{11}}(0)&\widehat{g_{12}}(n_*)\\\overline{\widehat{g_{12}}(n_*)}&-\mathrm {i}\widehat{g_{11}}(0)\end{pmatrix}M+Z(\theta )^{-1}\mathcal {R}_g(\theta )Z(\theta ):=\mathfrak {p}+\mathfrak {q}. \) 
Let \(e^{\mathfrak {p}+\mathfrak {q}}=e^{\mathfrak {p}}e^{f_+}\) , \(A_+=e^{2\pi \left( \xi -\frac{\langle n_*,\theta \rangle }{2}\right)J}e^\mathfrak {p}:=e^{a_+}\) , and \( B:=P\cdot e^Y\cdot Z \) , the system \( (\alpha ,Ae^f) \)  is conjugate to \( (\alpha ,A_+e^{f_+}) \)  by \( B \) .
Now we get the corresponding estimates. By the decay property of Fourier coefficients of \( g(\theta )\) , for any \( n\in {\mathbb {Z}}^d \) , we have
\(|\widehat{g_{11}}(n)|,|\widehat{g_{12}}(n)|\le 2\tilde{\epsilon } e^{- r(2\pi |n|)^\nu }.\) 
Then by the BCH Formula (REF ), one has
\(\Vert a_+\Vert &\le 2\left(\epsilon ^{\sigma }+4\epsilon ^{\frac{8}{9}}(1+e^{- r(2\pi |n_*|)^\nu })\right)\le 4\epsilon ^{\sigma },\\|f_+|_{r_+}&\le 2\cdot 4e^{ 2r_+(\pi |n_*|)^\nu }\cdot 2\epsilon ^{\frac{8}{9}} e^{-(r-r_+) (2\pi N_2)^\nu }\\&\quad \qquad \le \epsilon ^{\frac{3}{4}-\frac{2 r_+}{r-r_+}} e^{-(r-r_+) \epsilon ^{-\frac{10\nu }{11}\frac{\sigma }{\tau }}}\ll \epsilon ^{2},\\|B|_{r^{\prime \prime }}&\le 32 \gamma ^{-\frac{1}{2}}\Vert A\Vert ^{\frac{1}{2}} |n_*|^{\frac{\tau }{2}}e^{r^{\prime \prime }(\pi |n_*|)^{\nu }},\quad \forall r^{\prime \prime }\in (0, r_+].\) 

It is easy to see that \( \deg B=n_* \) , and (), thus we finish the proof of Proposition REF .

KAM scheme
Now we consider the initial cocycle \( (\alpha ,A_0e^{f_0}) \) . Denote its rotation number by \( \rho (\alpha ,A_0e^{f_0}) \) , Eliasson [2]} proved that if \( \rho (\alpha ,A_0e^{f_0}) \)  is Diophantine with respect to \( \alpha \) , that is, there exist \( \kappa ^{-1}>1 \) , \( \tau >d \)  such that \( \rho \in \mathrm {DC}_\alpha ^d(\kappa ,\tau ) \) , where
\(\mathrm {DC}_\alpha ^d(\kappa ,\tau )=\lbrace \phi \in {\mathbb {R}}^d: |2\phi -\langle m,\alpha \rangle |_{\ge \frac{\kappa }{(1+|m|)^\tau } \rbrace ,or \textit {rational} with respect to  \alpha  (i.e.,  2\rho (\alpha ,A_0e^{f_0})\equiv \langle k, \alpha \rangle \mod {{\mathbb {Z}}} for some  k\in {\mathbb {Z}}^{d} \backslash \lbrace 0\rbrace ), then the system  (\alpha ,A_0e^{f_0})  is reducible, thus is full measure reducible, for small analytic potentials. Here we will focus on a quantitative version of full measure reducibility.\begin{theorem}Given  r_0>0 . Assume  \alpha \in \mathrm {DC}_d(\gamma ,\tau ),  A_0\in \mathrm {SL}(2,{\mathbb {R}}) , f_0\in G^\nu _{r_0}(\mathbb {T}^d,\mathrm {sl(2,{\mathbb {R}})}), with 0<\nu <1. Then for any  r\in (0,r_0) , there exists \bar{c}=\bar{c}(\gamma ,\tau ,d,\nu )>0  and a numerical constant  \bar{D} , such that if\begin{equation}|f_0|_{r_0}=\epsilon _0<\epsilon _*:= \bar{c}\Vert A_0 \Vert ^{-\bar{D}}(r_0-r)^{\bar{D}\nu \tau },\end{equation}the following holds:\begin{enumerate}[label=\mathrm {()}\ \ ,align=left,widest=ii,labelsep=0pt,leftmargin=0.5em]\item The system  (\alpha ,A_0e^{f_0})  is strong almost reducible with the ``width^{\prime \prime } r.\item If  \rho (\alpha ,A_0e^{f_0})  is rational w.r.t. \alpha , then there exist  \widetilde{Z}\in G^\nu _{r}(2\mathbb {T}^d,\mathrm {SL}(2,{\mathbb {R}}))  and  A\in \mathrm {SL}(2,{\mathbb {R}})  such that\begin{equation}\widetilde{Z}(\theta +\alpha )^{-1}A_0e^{f_0(\theta )}\widetilde{Z}(\theta )=A,\end{equation}with estimates|\widetilde{Z}|_{r^{\prime \prime }}\le D_1|k|^{\tau }e^{2r^{\prime \prime }(\pi |k|)^{{\nu }}} , \forall r^{\prime \prime }\in (0,r],\ D_1=D_1(\gamma ,\tau ,\Vert A_0\Vert ,r_0,d)>0.Moreover, if  (\alpha ,A_0e^{f_0})  is not uniformly hyperbolic,then  A  can be chosen as\begin{pmatrix}1 &\varphi \\0 & 1\end{pmatrix} \text{ with } |\varphi |\le \epsilon _{0}^{\frac{3}{5}} e^{- r|2\pi k|^{{\nu }}}.\end{enumerate}\item If  \rho (\alpha ,A_0e^{f_0})\in \bigcup _{\gamma ^{\prime }>0}\mathrm {DC}_\alpha ^d(\gamma ^{\prime },\tau ) ,then there exist  B\in G^\nu _r(2d,\mathrm {SL}(2,{\mathbb {R}})) ,  A\in \mathrm {SL}(2,{\mathbb {R}})  such that\begin{equation}B(\theta +\alpha )^{-1}A_0e^{f_0(\theta )}B(\theta )=A, \quad \rho (\alpha ,A)\ne 0,\end{equation}with estimates  |B|_{r}\le \Gamma _1(\gamma ^{\prime },\gamma ,\tau ,d,\nu ,r,r_0,\Vert A_0\Vert ) ,  |\deg B|\le \Gamma _2(\gamma ^{\prime },\gamma ,\tau ,d,\nu ,r,r_0,\Vert A_0\Vert ) .\end{theorem}}\) 
We prove (i) by induction.
1). Take \(r_0,r,\epsilon _*\)  such that the first step holds.
2). Assume that we are at the \( (j+1)^{\mathrm {th}} \)  KAM step, where we have \( A_j\in \mathrm {SL}(2,{\mathbb {R}}) \)  with eigen-values \( e^{\pm \mathrm {i}2\pi \xi _j} \)  and \( f_j\in G^\nu _{r_j} \)  is satisfied \( |f_j|_{r_j}\le \epsilon _j \)  for some \( \epsilon _j\le \epsilon _0 \) . Let \( \tilde{r} =\frac{r_0+r}{2}\) . Then we define
\(r_j-r_{j+1}:=\frac{r_0-\tilde{r}}{4^{j+1}}, &\qquad &N_j:=\frac{1}{2\pi }\left(\frac{|\ln \epsilon _j|}{r_j-r_{j+1}}\right)^{\frac{1}{\nu }}=\frac{1}{2\pi }\left(\frac{4^{j+1}|\ln \epsilon _j|}{r_0-\tilde{r}}\right)^{\frac{1}{\nu }}.\) 

If \( \epsilon _j \)  is sufficiently small such that \(\epsilon =\epsilon _j\) , \(r=r_j\) , \(r_+=r_{j+1}\) , \(A=A_j\)  satisfying (REF ), then by the Proposition REF , there exist
\(Z_j\in G^\nu _{r_{j+1}}(2\mathbb {T}^d,\mathrm {SL}(2,{\mathbb {R}})), &\quad &A_{j+1}\in \mathrm {sl}(2,{\mathbb {R}}), &\quad &f_{j+1} \in G^\nu _{r_{j+1}}(\mathbb {T}^d,\mathrm {sl}(2,{\mathbb {R}})),\) 

such that
\( Z_j(\theta +\alpha )^{-1}A_je^{f_j(\theta )}Z_j(\theta )=A_{j+1}e^{f_{j+1}(\theta )}.\) 
Moreover,

If for any \( n\in {\mathbb {Z}}^d , 0<|n|\le N_j\) , we have \( |2\xi _j-\langle n,\alpha \rangle |_{\ge \epsilon ^{\frac{1}{20}} , then{\begin{@align}{3}{3}\Vert A_{j+1}-A_j\Vert \le 4\Vert A_j\Vert \epsilon _j, &\quad &|f_{j+1}|_{r_{j+1}}\le \epsilon _{j+1}:=\epsilon _j^2,&\quad & |Z_j-\mathrm {Id}|_{r_{j+1}}\le 2\epsilon _j^{\frac{1}{2}}.\end{@align}}\item If there is  n_j\in {\mathbb {Z}}^d ,  0<|n_j|\le N_j  such that  |2\xi _j-\langle n_j,\alpha \rangle |_{\le \epsilon ^{\frac{1}{20}},  then\begin{equation} A_{j+1}=e^{a_{j+1}}=M^{-1}\exp \begin{pmatrix}\mathrm {i}t_j & \mu _j\\\overline{\mu _j} & -\mathrm {i}t_j\end{pmatrix}M\end{equation}with estimates:\begin{equation}\begin{aligned}\Vert a_{j+1}\Vert &\le 4\epsilon _j^{\frac{1}{20}},\quad |t_j|\le \epsilon _j^{\frac{1}{20}},\quad |\mu _j|\le \epsilon _j^{\frac{3}{4}}e^{-r|2\pi n_*|^{\nu }}, \\|f_{j+1}|_{r_{j+1}}&\le \epsilon _{j+1}:=\epsilon _{j}^{\frac{3}{4}-\frac{2 r_{j+1}}{r_j-r_{j+1}}} e^{-(r_j-r_{j+1}) \epsilon _j^{-\frac{1}{22}\frac{\nu }{\tau }}}.\end{aligned}\end{equation}and\begin{equation}|Z_j|_{r^{\prime \prime }} \le 32\gamma ^{-\frac{1}{2}}\Vert A_j\Vert ^{\frac{1}{2}}|n_j|^{\frac{\tau }{2}} e^{r^{\prime \prime }(\pi |n_j|)^\nu }, \quad \forall r^{\prime \prime }\in (0,r_{j+1}].\end{equation}}}3). In view of (\ref {25}) and (\ref {26}) one sees that for any \)  j0 \(, \)  j02j \(, \)  Aj2A0 \(. So if \)  0<* \(, then Proposition \ref {prop} can be applied iteratively. Indeed, it suffices to consider (\ref {epsilon}) in each KAM step. \) j\( decays super-exponentially with \) j\(, while the right hand side of (\ref {epsilon}) decays exponentially with \) j\(.Hence \)  (, A0ef0) \( is almost reducible.\) For (ii). Assume that there are at least two resonant steps in the above procedure. We consider two consecutive resonant steps, assume they are the \( p^{\mathrm {th}} \)  and the \( q^{\mathrm {th}} \) . If at the \( q^{\mathrm {th}} \) , by the resonant condition, \( \left|2\xi _{ q} -\langle n_{q},\alpha \rangle \right|_{\le \epsilon _{q}^{\frac{1}{20}} , hence  \vert \xi _{ q}\vert >\frac{\gamma }{4|n_{q}|^\tau } . By the Proposition \ref {prop}, after the  p^{\mathrm {th}} , we have  \vert \xi _{ p+1}\vert \le 4\epsilon _{p}^{\frac{1}{20}}. By (\ref {25}) and (\ref {4.9}), we have  |\xi _{q} |\le 8\epsilon _{p}^{\frac{1}{20}}\le \frac{\epsilon _{p}^{\frac{1}{22}}\gamma }{4|n_{p}|^\tau }. Thus,\begin{equation}|n_{q}|\ge \epsilon _{p}^{-\frac{1}{22\tau }}|n_{p}|.\end{equation}If the  (j_m)^{\mathrm {th}}  is the resonant step, then  \deg Z_{j_m}=n_{j_m} . By (\ref {degree}) and (\ref {28}) and  2\rho (\alpha ,A_0e^{f_0})\equiv \langle k, \alpha \rangle \mod {{\mathbb {Z}}}, we have  \frac{\langle k,\alpha \rangle }{2}=\rho (\alpha ,A_{j_m+1}e^{f_{{j_m+1}}}) +\sum _{l=1}^m \frac{\langle n_{j_l},\alpha \rangle }{2},  then{\begin{@align*}{1}{-1}8\pi \epsilon _{j_m}^{\frac{1}{20}}\ge |2\pi \xi _{j_m+1}|&=|\rho (\alpha , A_{j_m+1}e^{f_{{j_m+1}}})|=\frac{1}{2}|\langle k-\sum _{l=1}^{m} n_{j_l},\alpha \rangle |\gtrsim \frac{1}{N_{j_m}^\tau },\end{@align*}}thus there are at most finitely many resonant steps.}Then by the estimates of \)  Zj \( in (\ref {25}) and the sequence \)  (rj)jN \( given in (\ref {24}), we can find that \)  j=0 Zj = ZGr(2Td, SL(2,R)) \( is well-defined, thus we have \)  Z(+)-1(A0ef0)Z=ea \( for some \)  asl(2,R) \( with \)  (,ea)=0 \(.\) Assume that there are \( s+1 \)  resonant steps totally, and the resonant points are
\( n_{j_0},\dots ,n_{j_s} \in {\mathbb {Z}}^d,\quad 0<|n_{j_i}|\le N_{j_i}, \, i=0,1,\dots ,s,\) 
then \( k= n_{j_0}+\cdots +n_{j_s} \) . By (), we have
\(\begin{aligned}|n_{j_s}|-\sum _{i=0}^{s-1}|n_{j_i}|&\le |k|\le |n_{j_s}|+\sum _{i=0}^{s-1}|n_{j_i}|,\\(1-2\epsilon _{j_{s-1}}^{\frac{1}{22\tau }})^\nu |n_{j_s}|^{\nu }&\le |k|^{\nu }\le (1+2\epsilon _{j_{s-1}}^{\frac{1}{22\tau }})^{\nu } |n_{j_s}|^{\nu },\\\prod _{i=0}^s \text{const.}|n_{j_i}|&\le (\text{const.}|n_{j_s}|)^2.\end{aligned}\) 
Then by (), for any \(r^{\prime \prime }\in (0,r],\)
\(|\widetilde{Z}|_{r^{\prime \prime }}&\le 4|Z_{j_0}|_{r^{\prime \prime }}\cdots |Z_{j_s}|_{r^{\prime \prime }}\\&\le 4\prod _{i=0}^s 32\gamma ^{-\frac{1}{2}}\Vert A_{j_i}\Vert ^{\frac{1}{2}}|n_{j_i}|^{\frac{\tau }{2}} e^{r^{\prime \prime }(\pi |n_{j_i}|)^{{\nu }}}\le D_1 |k|^{\tau }e^{2r^{\prime \prime }(\pi |k|)^{{\nu }}},\) 

where \(D_1=D_1(\gamma ,\tau ,\Vert A_0\Vert ,r_0,d)>0\) .
If \( (\alpha ,A_0e^{f_0}) \)  is not uniformly hyperbolic, which means \( a \)  can not be a hyperbolic matrix, thus we have \( \det a=0. \)  Assume \( a=\begin{pmatrix}a_{11}&a_{12}\\a_{21}&-a_{11}\end{pmatrix} \) . Then exists \( \vartheta \in \mathbb {T} \)  such that \( R_{-\vartheta }aR_\vartheta =\widetilde{a}:=\begin{pmatrix}0&a_{12}-a_{21}\\0&0\end{pmatrix} \) , replace \( \widetilde{Z} \)  by \( \widetilde{Z}R_\vartheta \) , and let \( \varphi =a_{12}-a_{21} \) , we get \( (\alpha ,A_0e^{f_0}) \)  is conjugated to \( (\alpha ,e^{\widetilde{a}}) \) .
Noting that if we rewrite \(a\)  as \(a=M^{-1}\begin{pmatrix}\mathrm {i}t&\mu \\\overline{\mu }&-\mathrm {i}t\end{pmatrix}M\)  with \(t\in {\mathbb {R}}\) , \(\mu \in \mathbb {C}\) , by (REF ), we have \(|\varphi |=2|t|\) . Now we consider after the last resonant step \( (\alpha ,A_{j_s+1}e^{f_{j_s+1}}) \) :
Clearly, we have
\(e^a=e^{a_{j_s+1}}\prod _{l=1}^{\infty }e^{\widehat{g_l}(0)},\) 
then apply the BCH Formula (REF ) again, we have
\(\Vert a_{j_s+1}-a \Vert \le 2\Vert \widehat{g_1}(0)\Vert \le 4\epsilon _{j_s+1}.\) 
By the structure of \(a_{j_s+1}\)  (c.f. ()), and \(\det a=0\) , we have
\(|t|=|\mu |\le \epsilon _{j_s}^{\frac{3}{4}} e^{- r_{j_s}(2\pi |n_{j_s}|)^\nu }+4\epsilon _{j_s+1}.\) 
Since \( \epsilon _{j_s+1}:=\epsilon _{j_s}^{\frac{3}{4}-\frac{2r_{j_s+1}}{r_{j_s}-r_{j_s+1}}} e^{-(r_{j_s}-r_{j_s+1}) \epsilon _{j_s}^{-\frac{10\nu }{11}\frac{\sigma }{\tau }}}\) , we get
\( |\varphi |\le 5\epsilon _{j_s}^{\frac{3}{4}} e^{- r_{j_s}(2\pi |n_{j_s}|)^{{\nu }}}\le \epsilon _{0}^{\frac{3}{5}}e^{-r|2\pi k|^{{\nu }}}.\) 
For (iii). Assume that we are at the \( (j+1)^{\mathrm {th}} \)  KAM step, that is, we have (REF ). Then set \( B_0=\mathrm {Id} \) , and let \( B_{j+1}=B_jZ_j(\theta ) \) , we have
\(B_{j+1}(\theta +\alpha )^{-1}A_0e^{f_0(\theta )}B_{j+1}(\theta )=A_{j+1}e^{f_{j+1}(\theta )},\) 
we distinguish two cases:
Non-resonant case: By \( 1^\circ \) , , we have
\(|B_{j+1}|_{r_{j+1}}\le (1+2\epsilon _j^{\frac{1}{2}})|B_{j}|_{r_j}, \text{ and } \deg B_{j+1}=\deg B_j.\) 
Resonant case: By \( 2^\circ \) , review Proposition REF , we have
\(\begin{aligned}|B_{j+1}|_{r_{j+1}}&\le 32\gamma ^{-\frac{1}{2}}\Vert A_j\Vert ^{\frac{1}{2}}|n_j|^{\frac{\tau }{2}}e^{r_{j+1}(\pi |n_{j_i}|)^{{\nu }}}|B_{j}|_{r_j}, \text{ and } \deg B_{j+1}=\deg B_j+n_j.\end{aligned}\) 
Thus we have
\(\begin{aligned}|\deg B_j|&\le \sum _{i=0}^{j-1} N_i\le \sum _{i=0}^{j-1}\frac{1}{2\pi }\left(\frac{4^{j+1}|\ln \epsilon _j|}{r_0-\tilde{r}}\right)^{\frac{1}{\nu }}\le \frac{|\ln \epsilon _j|^3}{r_0-\tilde{r}}.\end{aligned}\) 
Since \( \rho (\alpha ,A_0e^{f_0(\theta )})\in \mathrm {DC}_{\alpha }^d(\gamma ^{\prime },\tau ) \)  with some \( \gamma ^{\prime }>0 \) , for any \( m\in {\mathbb {Z}}^d \) , we have
\(\begin{aligned}|2\rho (\alpha ,A_je^{f_j(\theta )})-\langle m,\alpha \rangle |_{&=|2\rho (\alpha ,A_0e^{f_0(\theta )})+\langle \deg B_j,\alpha \rangle -\langle m,\alpha \rangle |_{\\&\ge \frac{\gamma ^{\prime }}{(|m-\deg B_j|+1)^\tau }\ge \frac{\gamma ^{\prime }(1+|\deg B_j|)^{-\tau }}{(|m|+1)^\tau }}which means  \rho (\alpha ,A_je^{f_j(\theta )})\in \mathrm {DC}_\alpha ^d(\gamma ^{\prime }(1+|\deg B_j|)^{-\tau },\tau ) .}Then we will need the following lemma which is essentially proved by Dinaburg-Sinai \cite {dinaburg1975one}, see also \cite {ge2019exponential}.\begin{lemma}Let  \rho (\alpha ,A_je^{f_j(\theta )})\in \mathrm {DC}_\alpha ^d(\kappa ,\tau ) , then if\begin{equation} |f_j|_{r_j}\le \epsilon _j< c\kappa ^{40\nu }\Vert A \Vert ^{-D}(r_j-r_{j+1})^{D\nu \tau },\end{equation}there exist  \bar{Z}\in G^\nu _{r}(d,\mathrm {SL}(2,{\mathbb {R}})) ,  A\in \mathrm {SL}(2,{\mathbb {R}})  such that\bar{Z}(\theta +\alpha )^{-1}A_je^{f_j(\theta )}\bar{Z}(\theta )=A,with estimates  |\bar{Z}-\mathrm {Id}|_{r}\le 4\epsilon _j^{\frac{1}{2}} ,  \Vert A-A_j\Vert \le 8\Vert A_0\Vert \epsilon _j .\end{lemma}{\begin{xmlelement*}{proof}By (\ref {lem4.3}), (\ref {degree}) and  \rho (\alpha ,A_je^{f_j(\theta )})\in \mathrm {DC}_\alpha ^d(\kappa ,\tau ) , for any  0<|n|\le N_j , we have\begin{aligned}|2\pi \xi _j-\langle n,\alpha \rangle |_\ge |\rho (\alpha ,A_je^{f_j(\theta )})-\langle n,\alpha \rangle |_{-|2\pi \xi _j-\rho (\alpha ,A_je^{f_j(\theta )})|\\&\ge \frac{\kappa }{(1+|n|)^{\tau }}-\epsilon _j\ge \frac{\kappa }{(1+N_j)^{\tau }}-\epsilon _j \ge \epsilon _j^{\frac{1}{20}}.}Thus there are no resonant steps, then by (\ref {25}), there exists  \bar{Z}(\theta ):=\prod _{j=0}^{\infty }Z_j(\theta )\in G_r^\nu (d,\mathrm {SL}(2,{\mathbb {R}}))  such that\bar{Z}(\theta +\alpha )^{-1}A_je^{f_j(\theta )}\bar{Z}(\theta )=A,with  | \bar{Z}-\mathrm {Id}|_{r}\le 4\epsilon _j^{\frac{1}{2}} ,  \Vert A-A_j\Vert \le 8\Vert A_0\Vert \epsilon _j .\end{aligned}\end{xmlelement*}Notice that  \epsilon _j\le e_0^{2^j} , one can select  j_0\in {\mathbb {Z}} to be the smallest integer such that\epsilon _j\le c(\frac{\gamma ^{\prime }}{(1+|\deg B_j|)^{\tau }})^{40\nu }\Vert A \Vert ^{-D}(r_j-r_{j+1})^{D\nu \tau },then by Lemma \ref {DinaburgSinai}, there exists  \bar{Z}\in G_r^\nu (d,\mathrm {SL}(2,{\mathbb {R}}))  such that\bar{Z}(\theta +\alpha )^{-1}B_{j_0}(\theta +\alpha )^{-1}A_0e^{f_0(\theta )}B_{j_0}(\theta )\bar{Z}(\theta )=A.Let  B=B_{j_0}\bar{Z}(\theta ) , by (\ref {28}), any two consecutive resonant steps separate exponentially, we have|B|_r\le 2|B_{j_0}|_{r_{j_0}}\le D_1 |n_{j_0}|^\tau e^{2r_0|\pi n_{j_0}|^\nu }\le \Gamma _1(\gamma ^{\prime },\gamma ,\tau ,d,\nu ,r,r_0,\Vert A_0\Vert ),|\deg B|=|\deg B_{j_0}|\le \frac{|\ln \epsilon _{j_0}|^3}{r_0-\tilde{r}}\le \Gamma _2(\gamma ^{\prime },\gamma ,\tau ,d,\nu ,r,r_0,\Vert A_0\Vert ).Notice that  \rho (\alpha ,A)\ne 0 , otherwise it will contradict to  \rho (\alpha ,A_0e^{f_0(\theta )})\in \mathrm {DC}_\alpha ^d(\gamma ^{\prime },\tau ) .}\section {Proof of Theorem \ref {thmlong}}A solution  u_E^\phi (n)  of  L_{\lambda ^{-1}v,\alpha ,\phi }u_E^\phi (n)=E u_E^\phi (n)  is called a \textit {normalized eigenfunction} of  L_{\lambda ^{-1}v,\alpha ,\phi }  if  \sum _n |u_E^\phi (n)|^2=1 .\begin{Definition}For any fixed  N\in {\mathbb {N}},  C>0 ,  \varepsilon >0 , a normalized eigenfunction  u_E^{\phi }(n)  of  L_{\lambda ^{-1}v,\alpha ,\phi }  is said to be  (\nu ,N,C,\varepsilon )-good if  |u_E^{\phi }(n)|\le e^{-C\varepsilon |n|^{\nu }}  for  |n|\ge (1-\varepsilon )N .\end{Definition}We denote by  u_j^\phi (n)  the  (\nu ,N,C,\varepsilon )-good eigenfunctions of  L_{\lambda ^{-1}v,\alpha ,\phi }  and by  E_j^\phi  the corresponding eigenvalues. Let\mathcal {E}_{\nu ,N,C,\varepsilon }^\phi =\lbrace E_j^\phi :u_j^\phi (n) \text{ is a } (\nu ,N,C,\varepsilon )\text{-good eigenfunction of } L_{\lambda ^{-1}v,\alpha ,\phi } \rbrace .Then we have the following criterion.\begin{proposition}For a.e.  \phi \in , suppose that there exists  C>0 , and for any  \delta >0 , there exist  N_0(\delta )>0  and  0<\varepsilon (\delta ,N_0,C,\nu )<1 , such that\begin{equation}\sharp \mathcal {E}_{\nu ,N,C,\varepsilon }^\phi \ge (1-\delta )(2N)^d\end{equation}holds for any  N\ge N_0(\delta )  and  \varepsilon =\varepsilon (\delta ,N_0,C,\nu ) . Then  L_{\lambda ^{-1}v,\alpha ,\phi }  has pure point spectrum with sub-exponentially for a.e.  \phi \in .\end{proposition}{\begin{xmlelement*}{proof}The proof are essentially contained in \cite {avila2017sharp}, we give the proof here for completeness.\end{xmlelement*}Let  P^\phi (\bigtriangleup )  be the spectral projection of  L_{\lambda ^{-1}v,\alpha ,\phi }  on to the set  \bigtriangleup , then for any  \delta _n\in \ell ^2({\mathbb {Z}}) , it induces the spectral measure:\langle P_{\bigtriangleup }^\phi \delta _n,\delta _n\rangle =\mu _{\delta _n,\phi }(\bigtriangleup ).Now we fix  \phi \in  such that (\ref {sharp1}) is satisfied. Denote thatK_{\nu ,N,C,\varepsilon }^\phi =\lbrace j\in {\mathbb {N}}:u_j^\phi (n) \text{ is a } (\nu ,N,C,\varepsilon )\text{-good eigenfunction of } L_{\lambda ^{-1}v,\alpha ,\phi }\rbrace .Notice that  \sharp K_{\nu ,N,C,\varepsilon }^\phi  is finite for any fixed  \nu ,N,C,\varepsilon , and pick  N\ge \max \lbrace N_0(\delta ),\frac{|n_0|}{\varepsilon }\rbrace  , one has\sum _{|n-n_0|\le N}|u_j^\phi (n)|^2>\sum _{|n|\le N(1-\varepsilon )}|u_j^\phi (n)|^2>1-\mathcal {C}(C,\varepsilon ,\nu ,d)2^d N^{d-1}e^{-C\varepsilon N^\nu (1-\varepsilon )^\nu },for  (\nu ,N,C,\varepsilon )-good eigenfunction  u_j^\phi (n) .}Let  \mathcal {E}_{C}(\phi )=\bigcup _{\varepsilon >0}\bigcup _{N>0}\mathcal {E}_{N,C,\varepsilon }^\phi . We have\begin{aligned}\frac{1}{(2N)^d}\sum _{|n-n_0|\le N}|\mu _{\delta _n,\phi }(\mathcal {E}_{C}(\phi ))|&> \frac{1}{(2N)^d}\sum _{|n|\le N(1-\varepsilon )}|\langle P_{\mathcal {E}_{N,C,\varepsilon }^\phi }^\phi \delta _n,\delta _n\rangle |\\&=\frac{1}{(2N)^d}\sum _{|n|\le N(1-\varepsilon )}\sum _{j\in K_{\nu ,N,C,\varepsilon }^\phi }|u_j^\phi (n)|^2\\&>\frac{\sharp K_{\nu ,N,C,\varepsilon }^\phi }{(2N)^d}(1-\mathcal {C}(C,\varepsilon ,\nu ,d)2^d N^{d-1}e^{-C\varepsilon N^\nu (1-\varepsilon )^\nu })\\&>(1-\delta )(1-\mathcal {C}(C,\varepsilon ,\nu ,d)2^d N^{d-1}e^{-C\varepsilon N^\nu (1-\varepsilon )^\nu }).\end{aligned}Since  \mathcal {E}_{C}(\phi )=\mathcal {E}_{C}(\phi +\langle n,\alpha \rangle ) , and notice that  \langle P_{\bigtriangleup }^\phi \delta _n,\delta _n\rangle =\langle P_{\bigtriangleup }^{\phi +\langle m,\alpha \rangle } \delta _{n+m},\delta _{n+m}\rangle , we can rewrite the inequalities above as\frac{1}{(2N)^d}\sum _{|n|\le N}|\mu _{\delta _{n_0},\phi +\langle n,\alpha \rangle }(\mathcal {E}_{C}(\phi ))|\ge (1-\delta )(1-\mathcal {C}(C,\varepsilon ,\nu ,d)2^d N^{d-1}e^{-C\varepsilon N^\nu (1-\varepsilon )^\nu }).Let  N\rightarrow \infty , and then let  \delta \rightarrow 0 , we have\int _{|\mu _{\delta _{n_0},\phi }(\mathcal {E}_{C}(\phi ))|\mathrm {d}\phi =1by the ergodic theorem of  {\mathbb {Z}}^d  actions \cite {schmidt1996ergodic}. It follows that  |\mu _{\delta _{n_0},\phi }(\mathcal {E}_{C}(\phi ))|=1  for a.e.  \phi , for arbitrary fixed  n_0\in {\mathbb {Z}}^d . Note that  u_j^\phi (n)  decays sub-exponentially if  E_j^\phi \in \mathcal {E}_{C}(\phi ) , which completes the proof.}\end{aligned}Denote \) 
={: DCd(,)} \(. Note that \)  |>0|=1 \( implies that for any \)  >0 \(, there exists \)  0 \( such that if \)  |'|0 \(, then \)  |'|1-3 \(. By ergodicity of rigid irrational rotations, we have\)\(\lim _{N\rightarrow \infty }\frac{1}{(2N)^d}\sum _{|n|\le N}\chi _{\Theta _{\gamma ^{\prime }}}(\phi +\langle n,\alpha \rangle )=\int _{\chi _{\Theta _{\gamma ^{\prime }}}(\phi )\mathrm {d}\phi =|\Theta _{\gamma ^{\prime }}|\ge 1-\frac{\delta }{3}.Thus we have for a.e.  \phi \in ,\sharp \lbrace n:\phi +\langle n,\alpha \rangle \in \Theta _{\gamma ^{\prime }},|n|\le N(1-\frac{\delta }{3})\rbrace \ge (1-\delta )(2N)^dholds for  N\ge N_0(\delta ) .}Then we show that for any \)  NN0() \(, any \)  (,SEm-1v)=+m,DCd(',) \( with \)  |m|N(1-3) \( can produce \)  (,N,C,) \(-good eigenfunctions. Consequently, we have (\ref {sharp1}) holds, then we can conclude Theorem \ref {thmlong}.\)
Suppose that \( \rho (\alpha ,S_{E_m}^{\lambda ^{-1}v})=\phi +\langle m,\alpha \rangle \in \mathrm {DC}_\alpha ^d(\gamma ^{\prime },\tau )\) , then by Theorem REF , for any \( r\in (0,r_0) \) , when \( \lambda >\lambda _0(\alpha ,d,\nu ,r_0,r)\) , there exists \( B_m\in G_r^\nu (2d,\mathrm {SL}(2,) \)  (by adding a unitary) such that
\(B_m(\theta +\alpha )^{-1}S_{E_m}^{\lambda ^{-1}v}B_m(\theta )=\begin{pmatrix}e^{\mathrm {i}2\pi (\phi +\langle m^{\prime },\alpha \rangle )}&c\\0&e^{-\mathrm {i}2\pi (\phi +\langle m^{\prime },\alpha \rangle )}\end{pmatrix},\) 
with estimates (omit other dependencies)
\(|B_m|_r\le C_1(\gamma ^{\prime }), \quad |m-m^{\prime }|\le 2|\deg B_m|\le C_2(\gamma ^{\prime }).\) 
Let \( B_m(\theta )=\begin{pmatrix}b_{11}(\theta )& b_{12}(\theta )\\b_{21}(\theta )& b_{22}(\theta )\end{pmatrix} \) , a direct calculation shows that
\((E-\lambda ^{-1}v)b_{11}(\theta )=b_{11}(\theta -\alpha )e^{-\mathrm {i}2\pi (\phi +\langle m^{\prime },\alpha \rangle )}+b_{11}(\theta +\alpha )e^{\mathrm {i}2\pi (\phi +\langle m^{\prime },\alpha \rangle )},\) 
we denote \( z_{11}(\theta )=e^{\mathrm {i}2\pi \langle m^{\prime },\theta \rangle }b_{11}(\theta ) \) , then one has
\( (E-\lambda ^{-1}v)z_{11}(\theta )=z_{11}(\theta -\alpha )e^{-\mathrm {i}2\pi \phi }+z_{11}(\theta +\alpha )e^{\mathrm {i}2\pi \phi }.\) 
Taking the Fourier transformation of (REF ), we have
\(\sum _{k\in {\mathbb {Z}}^d}\hat{z}_{11}(n-k)\hat{v}_k+2\lambda \cos 2\pi (\phi +\langle n,\alpha \rangle )\hat{z}_{11}(n)=\lambda E \hat{z}_{11}(n),\) 
i.e., \( \lbrace \hat{z}_{11}(n)\rbrace _{n\in {\mathbb {Z}}^d} \)  is an eigenfunction of the long range operator \( L_{\lambda ^{-1}v,\alpha ,\phi } \) . By the fact that \( \det |B_m(\theta )|=1 \) , we have
\(2\Vert b_{11}\Vert _{L^2}=\Vert b_{11}\Vert _{L^2}+\Vert b_{21}\Vert _{L^2}\ge \Vert B_m\Vert _{0}^{-1},\) 
which implies \( \Vert \hat{z}_{11}\Vert _{\ell ^2}=\Vert \hat{b}_{11}\Vert _{\ell ^2}=\Vert b_{11}\Vert _{L^2}\ge (2\Vert B_m\Vert _0^{-1})\) . Take \( u_{m}^{\phi }(n)=\frac{\hat{z}_{11}(n)}{\Vert \hat{z}_{11}\Vert _{\ell ^2} } \) , then it is a normalized eigenfunction of \( L_{\lambda ^{-1}v,\alpha ,\phi } \) . Let
\(2\varepsilon <1-(1-\frac{\delta }{3})^\nu -\frac{2\ln C_1}{r|2\pi |^\nu N^\nu }-\frac{|C_2|^\nu }{N^\nu }.\) 
Since \( u_{m}^{\phi }(n)=u_{m}^{\phi +\langle m^{\prime },\alpha \rangle }(n-m^{\prime }) \)  and the subadditivity of \( f(x)=x^\nu \) , then we have
\(\begin{aligned}|u_{m}^{\phi }(n)|&=|u_{m}^{\phi +\langle m^{\prime },\alpha \rangle }(n-m^{\prime })|\\&\le |B_m|_{r}^2 e^{-r|2\pi (n-m^{\prime })|^\nu }\\&\le C_1^2e^{-r|2\pi |^{\nu }(\sum _{i=1}^d|n_i-m^{\prime }_i|)^\nu }\\&\le C_1^2e^{-r|2\pi |^{\nu }(\sum _{i=1}^d|n_i|-|m^{\prime }_i|)^\nu }\\&\le C_1^2e^{-r|2\pi |^{\nu }(|n|^\nu -|m^{\prime }|^\nu )}\\&\le C_1^2e^{r|2\pi |^{\nu }(|m|+|m-m^{\prime }|)^\nu }e^{-r|2\pi n|^\nu }\\&\le e^{r|2\pi |^{\nu } (\frac{2\ln C_1}{r|2\pi |^{\nu } }+|m|^{\nu }+ |C_2|^\nu )}e^{-r|2\pi n|^\nu }\\&\le e^{|2\pi |^\nu r\varepsilon |n|^{\nu }},\end{aligned}\) 
for \( |n|\ge N(1-\delta ) \) , which means \( \lbrace u_{m}^{\phi }(n)\rbrace _{n\in {\mathbb {Z}}^d} \)  is \( (\nu ,N,|2\pi |^\nu r,\varepsilon ) \) -good. Notice that \( \varepsilon \)  can be chosen as \( \varepsilon (\delta ,N_0,C,\nu ) \) .
Proof of Theorem  REF
In this section, we prove Theorem REF  by the famous Moser-Pöschel argument of quasi-periodic Schrödinger cocycles and our reducibility results.
Rewrite the Schrödinger cocycle as
\(S_E^v=A_E+F(\theta ),\) 
where
\(A_E=\begin{pmatrix}E&-1\\1&0\end{pmatrix}, F(\theta )=\begin{pmatrix}-v&0\\0&0\end{pmatrix}.\) 
We only need to consider
\(E\in \Sigma _{v,\alpha }\subset [-2+\inf v,2+\sup v],\) 
since \({\mathbb {R}}\backslash \Sigma _{v,\alpha }\)  is uniformly hyperbolic. Thus the norm of \(A_E\)  is bounded uniformly with respect to \(E\) . Noting that
\(A_E+F(\theta )=A_EG=A_E\begin{pmatrix}1&0\\v&1\end{pmatrix},\) 
and \( \mathrm {tr} G=2 \) , by the exponential map, there exists \(f\in G^\nu _{r_0}(\mathbb {T}^d,\mathrm {sl}(2,{\mathbb {R}}))\)  such that
\(A_E+F=A_Ee^{f}, \text{ with } |f|_{r_0}\sim |v|_{r_0}.\) 
Fix \(r\in (0,r_0)\) , we consider the cocycle \( (\alpha , S^v_{E_k^+}) \) , where \( E_k^+\in \Sigma _{v,\alpha } \)  is the right edge point of gap \(G_k(v)\) , then we have
\( 2\rho (\alpha ,S_{E_k^+}^v)\equiv \langle k, \alpha \rangle \mod {{\mathbb {Z}}}\) , and \( (\alpha ,S_{E_k^+}^v) \)  is not uniformly hyperbolic. By Theorem REF , there exists \(\epsilon _*=\epsilon _*(\gamma ,\tau ,\nu ,r_0,r,d)>0\)  such that if \( |v|_{r_0}=\epsilon _0<\epsilon _*\) , then there exist \( Z\in G^\nu _{r}(2\mathbb {T}^d,\mathrm {SL}(2,{\mathbb {R}})) \) , \(c\in (0,1)\)  (\( c= 0 \)  iff it is a collapsed spectral gap, thus we need to do nothing), such that
\( Z(\cdot +\alpha )^{-1} S^v_{E_k^+}Z(\cdot )=B=\begin{pmatrix}1 &c \\0 & 1\end{pmatrix}.\) 
Noting that for any \(\delta >0\) , we have
\(Z(\cdot +\alpha )^{-1} S^v_{E_k^+-\delta }Z(\cdot )=B-\delta P(\cdot ),\) 
with
\( P(\cdot ) := \begin{pmatrix}z_{11}(\cdot )z_{12}(\cdot )-c z_{11}^2(\cdot )&-c z_{11}(\cdot )z_{12}(\cdot )+z^2_{12}(\cdot )\\-z_{11}^2(\cdot )&-z_{11}(\cdot )z_{12}(\cdot )\end{pmatrix}.\) 
For \( f:d\rightarrow , let \)  [f] :=d f() d\(. Using one step of averaging, one can construct a transformation \) Z\(, which conjugates the cocycle \)  (,B-P()) \( to \)  (,eb0-b1+2P1()) \( where \)\( b_0:=\begin{pmatrix}0&c\\0&0\end{pmatrix} ,\quad b_1:=\begin{pmatrix}[z_{11}z_{12}]-\frac{c}{2}[z_{11}^2]&-c[z_{11}z_{12}]+[z_{12}^2]\\-[z_{11}^2]&-[z_{11}z_{12}]+\frac{c}{2}[z_{11}^2]\end{pmatrix}. \)\(More precisely, we have the following lemma:\begin{lemma}For any R\in (0,r], let  D_R:=\sup _{n\in \mathbb {Z}^d} 4\gamma ^{-3}|n|^{3\tau }e^{-\frac{R}{2}(2\pi |n|)^\nu },  if \\ 0<\delta <4^{-1}D_R^{-1}|Z|_R^{-2} , then there exist  \widetilde{Z}\in G^\nu _{\frac{R}{2}}(\mathbb {T}^d,\mathrm {SL}(2,{\mathbb {R}}))  and  P_1\in G^\nu _{\frac{R}{2}}(\mathbb {T}^d,\mathrm {gl}(2,{\mathbb {R}})) such that\begin{equation}\widetilde{Z}(\cdot +\alpha )^{-1}(B-\delta P(\cdot ))\widetilde{Z}(\cdot )=e^{b_0-\delta b_1}+\delta ^2P_1(\cdot ),\end{equation}with the estimates\begin{equation}|\widetilde{ Z}-\mathrm {Id}|_{\frac{R}{2}}< 1,\quad |P_1|_{\frac{R}{2}}\le 8(2+D_R)^2|Z|_R^4+\delta ^{-1}c^2|Z|_R^2. \end{equation}\end{lemma}{\begin{xmlelement*}{proof}The proof follows a similar line as Lemma 6.1 in \cite {leguil2017asymptotics} (see also \cite {cai2021polynomial}), we will give the proof in Appendix \ref {appendixB} for completeness.\end{xmlelement*}}\)
Thus \( \widetilde{Z} \)  is homotopic to identity, we have
\( \rho (\alpha ,B-\delta P(\cdot ))=\rho (\alpha ,e^{b_0-\delta b_1}+\delta ^2P_1(\cdot )). \) 
In order to show \( |G_k(v)|\le \delta _1 \)  for some \( \delta _1>0 \) , by Gap Labelling Theorem, it suffices to show that \( \rho (\alpha , e^{b_0-\delta _1 b_1}+\delta _1^2P_1(\cdot )) >0 \) .
Let \( d(\delta ):=\det (b_0-\delta b_1) \) . By a direct computation, we get
\(d(\delta ) &=-\delta [z_{11}^2]c+\delta ^2([z_{11}^2][z_{12}^2]-[z_{11}z_{12}]^2)-\frac{\delta ^2}{4}c^2[z_{11}^2]^2 \\&=\delta ([z_{11}^2][z_{12}^2]-[z_{11}z_{12}]^2)\left( \delta -\frac{[z_{11}^2](c+\frac{1}{4}c^2\delta [z_{11}^2])}{[z_{11}^2][z_{12}^2]-[z_{11}z_{12}]^2}\right) .\) 
Let \(\tilde{r} =\frac{r_0+r}{2}, \chi =\frac{\tilde{r}-r}{6\tilde{r}}\) , and let \( \delta _1=c^{1-\chi } \) .
Claim 2 \( d(\delta _1)\ge \frac{9}{4}c^2 \) .

To get a lower bound of \(d(\delta _1)\) , we need to use the following lemma in [3]}.
Lemma 5.1 ([3]}) 
For any \( \kappa \in (0,\frac{1}{4}) \) , if \( c^{\frac{\kappa }{2}}|Z|_{d} \le \frac{1}{4} \) , we have
\(0<\frac{[z_{11}^2]}{[z_{11}^2][z_{12}^2]-[z_{11}z_{12}]^2}\le \frac{1}{2}c^{-\kappa },\quad [z_{11}^2][z_{12}^2]-[z_{11}z_{12}]^2\ge 8c^{2\kappa }.\) 

Reviewing (REF ), we get \(c\le \epsilon _{0}^{\frac{3}{5}}e^{-\frac{r}{1-\chi }|2\pi k|^{{\nu }}}\) , \(|Z|_{R}\le D^{\prime } e^{3R(\pi |k|)^{{\nu }}}\) .
Then we set \(R=\frac{1}{8}\frac{\chi }{1-\chi }r\) , for \(\epsilon _0\)  sufficient small, we have
\( 0<\delta _1\le \epsilon _0^{\frac{1}{2}}e^{-r|k|^{{\nu }}}<4^{-1}D_{R}^{-1}|Z|_R^{-2} .\) 
Hence we can apply Lemma , and conjugate the system to the cocycle \( (\alpha , e^{b_0-\delta _1 b_1}+\delta _1^2P_1(\cdot )) \) .
Since \( c^{\frac{\chi }{2}}|Z|_R\le \frac{1}{4} \) , then we can apply Lemma REF , and get
\(\frac{[z_{11}^2]c}{[z_{11}^2][z_{12}^2]-[z_{11}z_{12}]^2}\le \frac{1}{2}\delta _1,\) 
hence we have
\(d(\delta _1)\ge c^{1-\chi }\cdot 8c^{2\chi }\cdot c^{1-\chi }\left( \frac{1}{2}-\frac{1}{8}c^{2-\chi }[z_{11}^2]\right) \ge \frac{9}{4}c^2.\) 
By the expression of \( b_0 \)  and \( b_1 \) , we have
\(|b_0-\delta _1 b_1|\le c+\delta _1(1+c)|Z|^2_{R}\le \frac{3}{2}c^{1-\chi }|Z|_R^2.\) 
In view of Lemma 8.1 in [1]}, there exists \( \mathcal {P}\in \mathrm {SL}(2,{\mathbb {R}}) \)  with \( |\mathcal {P}|\le 2(\frac{|b_0-\delta _1 b_1|}{\sqrt{d(\delta _1)}})^{\frac{1}{2}} \) , such that
\( \mathcal {P}^{-1}e^{b_0-\delta _1 b_1}\mathcal {P}=R_{\sqrt{d(\delta _1)}}. \) 
Combining (REF ) and (REF ), we have
\(\frac{|b_0-\delta _1 b_1|}{\sqrt{d(\delta _1)}}\le \frac{\frac{3}{2}c^{1-\chi }|Z|_R^2}{\sqrt{\frac{9}{4}c^2}}\le c^{-\chi }|Z|_R^2.\) 
According to (REF ), and in view of (),
\(\begin{split}| \rho (\alpha , e^{b_0-\delta _1 b_1}+\delta _1^2P_1)-\sqrt{d(\delta _1)}| &\le \delta _1^2|\mathcal {P}|^2|P_1|_{\frac{R}{2}} \\&\le 32(2+D_R)^2 c^{2-3\chi }|Z|_R^6+4c^{3-2\chi }|Z|_R^2.\end{split}\) 
Since
\( 32(2+D_R)^2 c^{1-3\chi }|Z|_R^6+4c^{2-2\chi }|Z|_R^2<\frac{1}{2}, \) 
which implies that
\(\rho (\alpha , e^{b_0-\delta _1 b_1}+\delta _1^2P_1)\ge \sqrt{d(\delta _1)}-| \rho (\alpha , e^{b_0-\delta _1 b_1}+\delta _1^2P_1)-\sqrt{d(\delta _1)}| >0,\) 
this means \( |G_k(v)|\le \delta _1\le \epsilon _0^{\frac{1}{2}}e^{-r|2\pi k|^{{\nu }}} \) .

Proof of Theorem  REF
In this section, we will prove Theorem REF  by the following approach used in Takase [6]}.
It is known that the integrated density of states (IDS) is \( \frac{1}{2} \) -Hölder continuous in small coupling regime.
Theorem 6.1 ([7]}) 
Let \( \alpha \in \mathrm {DC}_d(\gamma ,\tau ) \) , \( v\in C^k(d,{\mathbb {R}}) \)  with \( k\ge 550\tau \) . Then there exists \( \epsilon _*=\epsilon _*(\gamma ,\tau ,d,k) \)  such that if \( |v|<\epsilon _* \) , then the integrated density of states \( N_{v,\alpha } \)  is \( \frac{1}{2}\) -Hölder continuous:
\(|N_{v,\alpha }(E+\epsilon )-N_{v,\alpha }(E-\epsilon )|\le C_0 \epsilon ^{\frac{1}{2}}, \ \forall \epsilon >0,\ \forall E\in {\mathbb {R}},\) 
where \( C_0 \)  depends only on \( \gamma ,\tau ,d \) .

With the help of Theorem REF , we have the following lemma.
Lemma 6.1 \( \lim _{|v_i|\rightarrow 0}\tau (\Sigma _i)=+\infty . \)

Suppose \( 0<|v_i|<\epsilon _0 \) , \( \epsilon _0 \)  sufficiently small, such that Theorem REF  and Theorem REF  hold. By the famous Gap-Labelling theorem, every bounded gap can be labelled by \( G_k(v)=[E_k^-,E_k^+] \)  with \( k\ne 0 \) . Without loss of generality, we only consider the bridge of \( \Sigma _i \)  at \( E_k^+ \) , denoted by \( C_k^+(v)=[E_k^+,F_k^+] \) , by the definition of bridge, \( F_k^+\in \Sigma _i \) . Thus
\(\tau (\Sigma ,E_k^+)=\frac{|F_k^+-E_k^+|}{|E_k^+-E_k^-|}.\) 
Since \( \Sigma _i \)  is a Cantor set with no isolated points, we have
\(E_k^+<F_k^+\le \sup \Sigma _i.\) 
Case 1. \( F_k^+=\sup \Sigma _i \) . Therefore
\(|N_{v,\alpha }(F_k^+)-N_{v,\alpha }(E_k^+)|=|\langle k,\alpha \rangle |_{{\begin{@align*}{1}{-1}\frac{|F_k^+-E_k^+|}{|E_k^+-E_k^-|}\ge \frac{(\frac{1}{C_0})^{2}|N_{v,\alpha }(F_k^+)-N_{v,\alpha }(E_k^+)|^2}{\epsilon _0^{\frac{1}{2}}e^{-r|2\pi k|^{{\nu }}}}\ge \frac{\gamma ^2}{C_0^2\epsilon _0^{\frac{1}{2}}}\frac{1}{e^{-r|2\pi k|^{{\nu }}}|k|^{2\tau }}.\end{@align*}}Case 2.  F_k^+<\sup \Sigma _i . Then  F_k^+  is the left boundary point of some nonempty gap  G_{k^{\prime }}(v) with  \ell (G_{k}(v))<\ell (G_{k^{\prime }}(v)) . For any nonempty gap  G_{k}(v) ,\zeta (k)=\left[\frac{1}{-r}\log [\epsilon _0^{-\frac{1}{2}}\ell (G_{k}(v))]\right]^{\frac{1}{\nu }}is well-defined. Thus by the spectral gap estimates,  \ell (G_{k}(v))\le \epsilon _0^{\frac{1}{2}}e^{-r|2\pi k|^{{\nu }}} , we have\zeta (k)\ge \zeta (k^{\prime }) \text{ and } \zeta (k)\ge \max \lbrace |k|,|k^{\prime }|\rbrace .Therefore|N_{v,\alpha }(F_k^+)-N_{v,\alpha }(E_k^+)|=|\langle k^{\prime }-k,\alpha \rangle |_{{\begin{@align*}{1}{-1}\frac{|F_k^+-E_k^+|}{|E_k^+-E_k^-|}\ge \frac{(\frac{1}{C_0})^{2}|N_{v,\alpha }(F_k^+)-N_{v,\alpha }(E_k^+)|^2}{\epsilon _0^{\frac{1}{2}}e^{-r\zeta (k)^{{\nu }}}}\ge \frac{\gamma ^2}{C_0^2\epsilon _0^{\frac{1}{2}}}\frac{1}{e^{-r\zeta (k)^{{\nu }}}|2\zeta (k)|^{2\tau }}.\end{@align*}}Notice that\tau (\Sigma )=\inf _u \tau (\Sigma ,u)=\inf _k \frac{|F_k^+-E_k^+|}{|E_k^+-E_k^-|}.Since  e^{-r\zeta (k)^{{\nu }}}|2\zeta (k)|^{2\tau }\le e^{-r|2\pi k|^{{\nu }}}|2k|^{2\tau }\rightarrow 0  when  k  large, thus there is a maximum  M  such that  e^{-r\zeta (k)^{{\nu }}}|2\zeta (k)|^{2\tau }\le M , then let  \epsilon _0\rightarrow 0 , we have\lim _{|v_i|\rightarrow 0}\tau (\Sigma _i)=+\infty .}We also need the following lemma in \cite {takase2021spectra}.\begin{lemma}[\cite {takase2021spectra}]Let  A:\mathcal {H}\rightarrow \mathcal {H}  and  B:\mathcal {H}\rightarrow \mathcal {H}  be bounded self-adjoint operators. Then|\mathrm {diam}\ \sigma (A)-\mathrm {diam}\ \sigma (B)|\le 2\mathrm {dist}_{\mathrm {Haus}}(\sigma (A),\sigma (B))\le 2\Vert A-B\Vert .\end{lemma}Now it suffices to check the conditions of Theorem \ref {astels}. For each set  \Sigma _i , by Lemma \ref {distHaus}, one has\lim \limits _{|v_i|\rightarrow 0}\mathrm {dist}_{\mathrm {Haus}}(\Sigma _i,[-2,2])=0 \text{ and } \lim \limits _{|v_i|\rightarrow 0}|\mathrm {diam}\Sigma _i-4|=0,and by Lemma \ref {thicklem},  \lim \limits _{|v_i|\rightarrow 0}\tau (\Sigma _i)=+\infty .Then there exists  \varepsilon =\varepsilon (\gamma ,\tau ,\nu ,r_0,d)>0 , such that  \sum \limits _{i=1}^b\frac{\tau (\Sigma _i)}{\tau (\Sigma _i)+1}\ge 1  and\begin{equation*}\left\lbrace \begin{aligned}&\Gamma (\Sigma _j)\le \mathrm {diam}(\Sigma _i),\ \ \forall 1\le j<i\le b,\\&\Gamma (\Sigma _i)\le \mathrm {diam}(\Sigma _1)+\cdots +\mathrm {diam}(\Sigma _{i-1}),\ \ \forall 2\le i\le b,\end{aligned}\right.\end{equation*}if  0<|v_i|<\varepsilon  for  i=1,\cdots ,b , then  \hat{\Sigma }=\Sigma _1+\Sigma _2+\cdots +\Sigma _b  is an interval.}\) 
Acknowledgments
We would like to thank Long Li, Qi Zhou and Hongyu Cheng for useful discussions, and Jiangong You for reading the earlier versions of the paper. This work is partially supported by National Key R&D Program of China (2020YFA0713300) and Nankai Zhide Foundation.

appendix

Proof of Lemma  REF
Here we prove Lemma REF  by the following quantitative Implicit Function Theorem:
Theorem 1.1 ([8]}) 
Let \( X,Y,Z \)  be Banach space, \( U\subset X \)  and \( V\subset Y \)  are neighborhoods of \( x_0 \)  and \( y_0 \)  respectively. Fix \( s,\delta >0 \)  and define \( X_s:=\lbrace x\in X ;\Vert x-x_0\Vert _X<s\rbrace  \)  and \( Y_\delta :=\lbrace y\in Y ;\Vert y-y_0\Vert _Y<\delta \rbrace  \) . Let \( \Psi \in C^{1}(U\times V,Z) \) . Suppose that \( \Psi (x_0,y_0)=0 \) , and that \( D_y\Psi (x_0,y_0)\in \mathcal {L}(Y,Z) \)  is invertible. If
\( \sup _{\overline{X_s}}\Vert \Psi (x,y_0)\Vert _Z\le \frac{\delta }{2\Vert D_y\Psi (x_0,y_0)^{-1}\Vert _{\mathcal {L}(Z,Y)}}, \) 
\( \sup _{\overline{X_s}\times \overline{Y_\delta }}\Vert \mathrm {Id}_Y-D_y\Psi (x_0,y_0)^{-1}D_y\Psi (x,y)\Vert _{\mathcal {L}(Y,Y)}\le \frac{1}{2}, \) 
then there exists \( y\in C^{1}(X_s, \overline{Y_\delta }) \) such that \( \Psi (x,y(x))=0. \)

We construct the nonlinear functional
\(\Psi :G_r^\nu (\mathbb {T},\mathrm {sl(2,{\mathbb {R}})})\times G_{r}^{\nu \,(nre)}(\eta ) \rightarrow G_{r}^{\nu \,(nre)}(\eta )\) 
by
\(\Psi (f,Y)=\mathbb {P}_{nre}[\ln \left( e^{-A^{-1}Y(\theta +\alpha )A}e^{f(\theta )}e^{Y(\theta )}\right) ],\) 
where \( \ln (\cdot ) \)  is the principal logarithm of a matrix.
One can easily check that
\( \Psi (0,0)=0,\ \Vert \Psi (f,0)\Vert \le |f|_{r}. \) 
By the definition of Fréchet derivative, We only need to consider the linear terms of \( \Psi (f,Y+Y^{\prime })-\Psi (f,Y) \) , we have
\(\Psi (f,Y+Y^{\prime }&)-\Psi (f,Y)=\mathbb {P}_{nre}\left[ \ln \left( e^{-A^{-1}(Y(\theta +\alpha )+Y^{\prime }(\theta +\alpha ))A}e^{f(\theta )}e^{Y(\theta )+Y^{\prime }(\theta +\alpha )}\right)\right.\\&\quad -\ln \left.\left( e^{-A^{-1}Y(\theta +\alpha )A}e^{f(\theta )}e^{Y(\theta )}\right)\right]\\&=\mathbb {P}_{nre}\left\lbrace \ln \left( e^{-A^{-1}(Y(\theta +\alpha )+Y^{\prime }(\theta +\alpha ))A}e^{f(\theta )}e^{Y(\theta )+Y^{\prime }(\theta )}\right)\right.\\&\quad -\ln \left( e^{-A^{-1}(Y(\theta +\alpha )+Y^{\prime }(\theta +\alpha ))A}e^{f(\theta )}e^{Y(\theta )}\right)\\&\quad +\ln \left( e^{-A^{-1}(Y(\theta +\alpha )+Y^{\prime }(\theta +\alpha ))A}e^{f(\theta )}e^{Y(\theta )}\right)\\&\quad -\ln \left.\left( e^{-A^{-1}Y(\theta +\alpha )A}e^{f(\theta )}e^{Y(\theta )}\right)\right\rbrace  .\) 

By the Baker-Campbell-Hausdorff (BCH) Formula
\( e^{X}e^Y= e^{X+Y+\frac{1}{2}[X,Y]+\frac{1}{12}([X,[X,Y]]+[Y,[Y,X]])+\cdots },\) 
a direct calculation shows that
\(D_y\Psi (f,Y)(Y^{\prime })&=\mathbb {P}_{nre}\lbrace -A^{-1}Y^{\prime }(\theta +\alpha )A+Y^{\prime }(\theta )+E+F\rbrace ,\) 
where \(E\)  is a sum of terms of at least 2 orders in \(-A^{-1}Y(\theta +\alpha )A\) , \(f\) , \(Y^{\prime }\) , and \(F\)  is a sum of terms of at least 2 orders in \(-A^{-1}Y^{\prime }(\theta +\alpha )A\) , \(f\) , \(Y\) , but both \(E\)  and \(F\)  are only 1 order in \(Y^{\prime }\) .
Let \( f=0,Y=0 \) , thus \(E, F\)  equal zero, then
\( D_y\Psi (0,0)(Y^{\prime })=\mathbb {P}_{nre}\lbrace -A^{-1}Y^{\prime }(\theta +\alpha )A+Y^{\prime }(\theta )\rbrace . \) 
By a direct computation, which is in analogy with Lemma 1 of [2]}, we have
\( \Vert D_y\Psi (0,0)(Y^{\prime })\Vert =|-A^{-1}Y^{\prime }(\theta +\alpha )A+Y^{\prime }(\theta )|_{r}\ge \tilde{\eta }|Y^{\prime }|_{r}, \) 
where
\(\tilde{\eta }:=\frac{\eta ^3}{\eta ^2+3\Vert A\Vert ^2\eta +2\Vert A\Vert ^4}.\) 
So we have
\( \Vert D_y\Psi (0,0)^{-1}\Vert \le \tilde{\eta }^{-1}. \) 
Assume that \( \eta \ge \Vert A\Vert ^2\epsilon ^\frac{1}{9} \) , then we have
\(\tilde{\eta }&=\frac{\eta ^3}{\eta ^2+3\Vert A\Vert ^2\eta +2\Vert A\Vert ^4}\\&\ge \frac{\Vert A\Vert ^2}{3\left(\frac{\Vert A\Vert ^2}{\eta }+(\frac{\Vert A\Vert ^2}{\eta })^2+(\frac{\Vert A\Vert ^2}{\eta })^3\right)}\\&\ge \frac{1}{9}\Vert A\Vert ^2\epsilon ^{\frac{1}{3}}.\) 

Set \( s=\epsilon ,\delta =\epsilon ^{\frac{1}{2}} \) , we have
\( 2\Vert D_y\Psi (0,0)^{-1}\Vert \cdot \sup _{\overline{X_s}}\Vert \Psi (x,0)\Vert  \le 2\tilde{\eta }^{-1}\epsilon \le \delta . \) 
On the other hand,
\(\Vert D_y\Psi (0,0)(Y^{\prime })-D_y\Psi (f,Y)(Y^{\prime })\Vert &=\Vert \mathbb {P}_{nre}\lbrace E+F\rbrace \Vert \\&\le \left( 8\Vert A\Vert ^2|Y|_{r}+8\Vert A\Vert ^2|Y|_{r}\right) |Y^{\prime }|_{r}\\&\le 16\Vert A\Vert ^2\epsilon ^{\frac{1}{2}}|Y^{\prime }|_{r},\) 

so if \(\epsilon \)  sufficiently small, we have
\(\Vert \mathrm {Id}_Y-D_y\Psi (0,0)^{-1}D_y\Psi (x,y)\Vert &\le \Vert D_y\Psi (0,0)^{-1}\Vert \Vert D_y\Psi (0,0)-D_y\Psi (x,y)\Vert \\&\le \tilde{\eta }^{-1}\times 16\Vert A\Vert ^2\epsilon ^{\frac{1}{2}}\le \frac{1}{2},\) 

By Theorem REF , for \( |f|_{r}\le \epsilon \) , and \( \eta \ge \Vert A\Vert ^2\epsilon ^{\frac{1}{9}} \) , there exists \( |Y|_{r}\le \epsilon ^{\frac{1}{2}} \)  such that \( \Psi (f,Y)=0 \) , i.e.
\( \ln \left( e^{-A^{-1}Y(\theta +\alpha )A}e^{f(\theta )}e^{Y(\theta )}\right)=f^{(re)}(\theta ), \) 
which is equivalent to
\( e^{-Y(\theta +\alpha )}(Ae^{f(\theta )})e^{Y(\theta )}=Ae^{f^{(re)}(\theta )}. \) 
It is easy to check \( |f^{(re)}|_{r}\le 2\epsilon \) , which complete the proof of Lemma REF .

Proof of Lemma 
Let \(G=-\delta B^{-1}P\) , we have \(\mathrm {tr}(B^{-1}P)=0\) , hence \(G\in G^\nu _{R}(\mathbb {T}^d,\mathrm {sl}(2,{\mathbb {R}}))\) . Then we can construct \(Y\in G^\nu _{\frac{R}{2}}(\mathbb {T}^d,\mathrm {sl}(2,{\mathbb {R}}))\)  (by identifying the Fourier coefficients) such that
\(B^{-1}Y(\cdot +\alpha )B-Y=G-[G].\) 
A direct calculation shows that
\(|\widehat{Y}(n)|\le 4\gamma ^{-3}|n|^{3\tau }|\widehat{G}(n)|,\) 
thus
\(|Y|_{\frac{R}{2}}&=\sum _{n\in \mathbb {Z}^d}|\widehat{Y}(n)|e^{\frac{R}{2}(2\pi |n|)^\nu }\\&\le \sum _{n\in \mathbb {Z}^d}4\gamma ^{-3}|n|^{3\tau }e^{-\frac{R}{2}(2\pi |n|)^\nu }|\widehat{G}(n)|e^{ R(2\pi |n|)^\nu }\le D_R\delta |P|_R.\) 
Then for \(\widetilde{Z}:=e^Y\) , we have
\(\widetilde{Z}(\cdot +\alpha )^{-1}(B-\delta P(\cdot ))\widetilde{Z}(\cdot )=(B-\delta [P])+\widetilde{P}(\cdot ),\) 
where
\(\widetilde{P}(\cdot ):=\sum _{m+n\ge 2}\frac{1}{m!}(-Y(\cdot +\alpha ))^m B\frac{1}{n!}Y(\cdot )^n-\delta \sum _{m+n\ge 1}\frac{1}{m!}(-Y(\cdot +\alpha ))^m P(\cdot )\frac{1}{n!}Y(\cdot )^n. \) 
Obviously, \(|\widetilde{ Z}-\mathrm {Id}|_{\frac{R}{2}}\le 2|Y|_{\frac{R}{2}}<1\) . Since \(\sum _{m+n=k}\frac{k!}{m!n!}=2^k\) , and \(|G|_R\le \delta |P|_R\) , we get
\(|\widetilde{P}(\cdot )|_{\frac{R}{2}}\le D_R^2\delta ^2|P|_R^2+4D_R\delta ^2|P|_R^2\le (2+D_R)^2\delta ^2|P|_R^2. \) 
By a direct calculation, we can see that
\(B-\delta [P]=I+(b_0-\delta b_1)+\frac{1}{2}(b_0-\delta b_1)^2-\delta ^2 \frac{1}{2} b_1^2.\) 
Let \(P_1:=-\frac{1}{2} b_1^2+\delta ^{-2}(-\sum _{k\ge 3}\frac{1}{k!}(b_0-\delta b_1)^k+\widetilde{P})\) , note that \(b_0\)  is nilpotent, a direct computation shows (), we finish the proof of Lemma .
